The classical approach (see [1]}) is to associate to \((\rho ,V)\)  a certain real vector bundle \(\mathcal {V}\)  over the classifying space \(BG\) . The Stifel Whitney class of \(\rho \)  is then defined to be the Stifel Whitney class of \(\mathcal {V}\)  as developed in [2]}.
Alternatively, the paper [3]} gives an axiomatic characterization of Stiefel Whitney classs of real representations. With the following modification we obtain a theory of Stiefel Whitney classs of orthogonal complex representations.
